Authors/Aristotle/priora/Liber 1/C23

Chapter 23

Greek	Latin	English
	(PL 64 0665A) CAPUT XXII/ XXIII. De syllogismo ostensivo.	23
40b17 Ὅτι μὲν οὖν οἱ ἐν τούτοις τοῖς σχήμασι συλλογισμοὶ τελειοῦνταί τε διὰ τῶν ἐν τῶι πρώτωι σχήματι καθόλου συλλογισμῶν καὶ εἰς τούτους ἀνάγονται, δῆλον ἐκ τῶν εἰ ρημένων· ὅτι δ᾽ ἁπλῶς πᾶς συλλογισμὸς οὕτως ἕξει, νῦν ἔσται φανερόν, ὅταν δειχθῆι πᾶς γινόμενος διὰ τούτων τινὸς τῶν σχημάτων.	(0665B) Quoniam igitur qui in his figuris sunt syllogismi perficiuntur per eos qui in prima figura sunt universales syllogismos, et in hos reducuntur, palam ex dictis. Quoniam autem simpliciter omnis syllogismus sic se habebit, nunc erit manifestum, cum ostensus fuerit omnis qui fit, per aliquam harum figurarum fieri.	It is clear from what has been said that the syllogisms in these figures are made perfect by means of universal syllogisms in the first figure and are reduced to them. That every syllogism without qualification can be so treated, will be clear presently, when it has been proved that every syllogism is formed through one or other of these figures.
Ἀνάγκη δὴ πᾶσαν ἀπόδειξεν καὶ πάντα συλλογισμὸν ἢ ὑπάρχον τι ἢ μὴ ὑπάρχον δεικνύναι, καὶ τοῦτο ἢ καθόλου ἢ κατὰ μέρος, ἔτι ἢ δεικτικῶς ἢ ἐξ ὑποθέσεως. τοῦ δ᾽ ἐξ ὑποθέσεως μέρος τὸ διὰ τοῦ ἀδυνάτου. πρῶτον οὖν εἴπωμεν περὶ τῶν δεικτικῶν· τούτων γὰρ δειχθέντων φανερὸν ἔσται καὶ ἐπὶ τῶν εἰς τὸ ἀδύνατον καὶ ὅλως τῶν ἐξ ὑποθέσεως.	Necesse est ergo omnem demonstrationem et omnem syllogismum aut inesse quid, aut non inesse monstrare. Et hoc aut universaliter, aut particulariter, amplius aut ostensive, aut ex hypothesi. Eius autem quod est ex hypothesi, pars est per impossibile. Primum ergo dicemus de ostensivis, his enim ostensis, manifestum erit et de iis qui ad impossibile, et omnino de iis qui ex hypothesi.	It is necessary that every demonstration and every syllogism should prove either that something belongs or that it does not, and this either universally or in part, and further either ostensively or hypothetically. One sort of hypothetical proof is the reductio ad impossibile. Let us speak first of ostensive syllogisms: for after these have been pointed out the truth of our contention will be clear with regard to those which are proved per impossibile, and in general hypothetically.
Εἰ δὴ δέοι τὸ Α κατὰ τοῦ Β συλλογίσασθαι ἢ ὑπάρ- χον ἢ μὴ ὑπάρχον, ἀνάγκη λαβεῖν τι κατά τινος. εἰ μὲν οὖν τὸ Α κατὰ τοῦ Β ληφθείη, τὸ ἐξ ἀρχῆς ἔσται εἰλημμένον. εἰ δὲ κατὰ τοῦ Γ, τὸ δὲ Γ κατὰ μηδενός, μηδ᾽ ἄλλο κατ᾽ ἐκείνου, μηδὲ κατὰ τοῦ Α ἕτερον, οὐδεὶς ἔσται συλλογισμός· τῶι γὰρ ἓν καθ᾽ ἑνὸς ληφθῆναι οὐδὲν συμβαίνει ἐξ ἀνάγκης. ὥστε προσληπτέον καὶ ἑτέραν πρότασιν. ἐὰν μὲν οὖν ληφθῆι τὸ Α κατ᾽ ἄλλου ἢ ἄλλο κατὰ τοῦ Α, ἢ κατὰ τοῦ Γ ἕτερον, εἶναι μὲν συλλογισμὸν οὐδὲν κωλύει, πρὸς μέντοι τὸ Β οὐκ ἔσται διὰ τῶν εἰλημμένων. οὐδ᾽ ὅταν τὸ Γ ἑτέρωι, κἀκεῖνο ἄλλωι, καὶ τοῦτο ἑτέρωι, μὴ ↵ συνάπτηι δὲ πρὸς τὸ Β, οὐδ᾽ οὕτως ἔσται πρὸς τὸ Β συλλογισμός.	Si ergo oporteat A de B syllogizare, vel inesse, vel non inesse, necesse est sumere aliquid de aliquo. (0665C) Si ergo A sumatur de B, quod ex principio erit sumptum, si autem A de C, C autem de nullo alio, nec aliud de illo C, neque de A alterum, neque de altero A, nullus erit syllogismus, nam in eo quod unum de uno sumitur, nihil accidit ex necessitate, quare assumenda est altera propositio. Si igitur sumatur A de alio, aut aliud de A, aut de C alterum, esse quidem syllogismum nihil prohibet, ad B autem non erit per ea quae sumpta sunt, nec quando C inest alteri, et illud alii, et hoc alteri, non copuletur autem ad B, nec sic erit ad B syllogismus ipsius A.	If then one wants to prove syllogistically A of B, either as an attribute of it or as not an attribute of it, one must assert something of something else. If now A should be asserted of B, the proposition originally in question will have been assumed. But if A should be asserted of C, but C should not be asserted of anything, nor anything of it, nor anything else of A, no syllogism will be possible. For nothing necessarily follows from the assertion of some one thing concerning some other single thing. Thus we must take another premiss as well. If then A be asserted of something else, or something else of A, or something different of C, nothing prevents a syllogism being formed, but it will not be in relation to B through the premisses taken. Nor when C belongs to something else, and that to something else and so on, no connexion however being made with B, will a syllogism be possible concerning A in its relation to B.
ὅλως γὰρ εἴπομεν ὅτι οὐδεὶς οὐδέποτε ἔσται συλλογισμὸς ἄλλου κατ᾽ ἄλλου μὴ ληφθέντος τινὸς μέσου, ὁ πρὸς ἑκάτερον ἔχει πως ταῖς κατηγορίαις·	Omnino enim dicimus quoniam nullus nunquam erit syllogismus alius de alio, non sumpto aliquo medio, quod ad utrumque se habet quoquo modo praedicationibus.	For in general we stated that no syllogism can establish the attribution of one thing to another, unless some middle term is taken, which is somehow related to each by way of predication.
ὁ μὲν γὰρ συλλογισμὸς ἁπλῶς ἐκ προτάσεών ἐστιν, ὁ δὲ πρὸς τόδε συλλογισμὸς ἐκ τῶν πρὸς τόδε προτάσεων, ὁ δὲ τοῦδε πρὸς τόδε διὰ τῶν τοῦδε πρὸς τόδε προτάσεων. ἀδύνατον δὲ πρὸς τὸ Β λαβεῖν πρότασιν μηδὲν μήτε κατηγοροῦντας αὐτοῦ μήτ᾽ ἀπαρνουμένους, ἢ πάλιν τοῦ Α πρὸς τὸ Β μη δὲν κοινὸν λαμβάνοντας ἀλλ᾽ ἑκατέρου ἴδια ἄττα κατηγοροῦντας ἢ ἀπαρνουμένους. ὥστε ληπτέον τι μέσον ἀμφοῖν, ὁ συνάψει τὰς κατηγορίας, εἴπερ ἔσται τοῦδε πρὸς τόδε συλλογισμός.	(0665D) Nam syllogismus quidem simpliciter ex propositionibus est, ad hoc autem syllogismus ex propositionibus, quae ad hoc, qui autem est huius ad hoc, per propositiones huius ad hoc, impossibile est autem ad B sumere propositionem, nihil neque praedicantes de eo, neque negantes, aut rursum eius quod est A ad B, nihil commune sumentes, sed utriusque propria quaedam praedicantes, aut negantes, quare sumendum, utriusque quod copulet praedicationes, si erit huius ad hoc syllogismus.	For the syllogism in general is made out of premisses, and a syllogism referring to this out of premisses with the same reference, and a syllogism relating this to that proceeds through premisses which relate this to that. But it is impossible to take a premiss in reference to B, if we neither affirm nor deny anything of it; or again to take a premiss relating A to B, if we take nothing common, but affirm or deny peculiar attributes of each. So we must take something midway between the two, which will connect the predications, if we are to have a syllogism relating this to that.
εἰ οὖν ἀνάγκη μέν τι λαβεῖν πρὸς ἄμφω κοινόν, τοῦτο δ᾽ ἐνδέχεται τριχῶς (ἢ γὰρ τὸ Α τοῦ Γ καὶ τὸ Γ τοῦ Β κατηγορήσαντας, ἢ τὸ Γ κατ᾽ ἀμφοῖν, ἢ ἄμφω κατὰ τοῦ Γ), ταῦτα δ᾽ ἐστὶ τὰ εἰρημένα σχήματα, φανερὸν ὅτι πάντα συλλογισμὸν ἀνάγκη γίνεσθαι διὰ τούτων τινὸς τῶν σχημάτων. ὁ γὰρ αὐτὸς λόγος καὶ εἰ διὰ πλειόνων συνάπτοι πρὸς τὸ Β· ταὐτὸ γὰρ ἔσται σχῆμα καὶ ἐπὶ τῶν πολλῶν. Ὅτι μὲν οὖν οἱ δεικτικοὶ περαίνονται διὰ τῶν προειρημένων σχημάτων, φανερόν·	Ergo si necesse est aliquod sumere ad utrumque commune, hoc autem contingit tripliciter, aut enim A de C et de B praedicantes, aut C de utrisque, aut utraque de C, hae autem sunt tres dictae figurae. Manifestum quoniam omnem syllogismum necesse est fieri per aliquam harum figurarum. Nam eadem ratio est, etsi per plura copuletur ad B, eadem enim erit figura et in pluribus. (0666A) Quoniam igitur ostensivi terminantur per praedictas figuras, manifestum est.	If then we must take something common in relation to both, and this is possible in three ways (either by predicating A of C, and C of B, or C of both, or both of C), and these are the figures of which we have spoken, it is clear that every syllogism must be made in one or other of these figures. The argument is the same if several middle terms should be necessary to establish the relation to B; for the figure will be the same whether there is one middle term or many. It is clear then that the ostensive syllogisms are effected by means of the aforesaid figures; these considerations will show that reductiones ad also are effected in the same way.
	(PL 64 0666A) CAPUT XXIII/ XXIV. De syllogismo ex hypothesi.
ὅτι δὲ καὶ οἱ εἰς τὸ ἀδύνατον, δῆλον ἔσται διὰ τούτων. πάντες γὰρ οἱ διὰ τοῦ ἀδυνάτου περαίνοντες τὸ μὲν ψεῦδος συλλογίζονται, τὸ δ᾽ ἐξ ἀρχῆς ἐξ ὑποθέσεως δεικνύουσιν, ὅταν ἀδύνατόν τι συμβαίνηι τῆς ἀντιφάσεως τεθείσης, οἷον ὅτι ἀσύμμετρος ἡ διάμετρος διὰ τὸ γί- νεσθαι τὰ περιττὰ ἴσα τοῖς ἀρτίοις συμμέτρου τεθείσης. τὸ μὲν οὖν ἴσα γίνεσθαι τὰ περιττὰ τοῖς ἀρτίοις συλλογίζεται, τὸ δ᾽ ἀσύμμετρον εἶναι τὴν διάμετρον ἐξ ὑποθέσεως δείκνυσιν, ἐπεὶ ψεῦδος συμβαίνει διὰ τὴν ἀντίφασιν. τοῦτο γὰρ ἦν τὸ διὰ τοῦ ἀδυνάτου συλλογίσασθαι, τὸ δεῖξαί τι ἀδύνατον διὰ τὴν ἐξ ἀρχῆς ὑπόθεσιν.	Quoniam autem et qui ad impossibile, palam erit per haec, omnes enim qui per impossibile concludunt, falsum quidem syllogizant. Quod autem ex principio erat, ex hypothesi demonstrant, quando aliquid accidit impossibile posita contradictione, ut quoniam diameter est asymeter, eo quod fiunt abundantia aequalia perfectis, posito symetro. Ergo aequalia quidem fieri abundantia perfectis syllogizant, asymetrum autem esse diametrum, ex hypothesi monstrant, quoniam falsum accidit propter contradictionem. (0666B) Hoc enim fuit per impossibile syllogizare, ostendere aliquid impossibile propter priorem hypothesin.	For all who effect an argument per impossibile infer syllogistically what is false, and prove the original conclusion hypothetically when something impossible results from the assumption of its contradictory; e.g. that the diagonal of the square is incommensurate with the side, because odd numbers are equal to evens if it is supposed to be commensurate. One infers syllogistically that odd numbers come out equal to evens, and one proves hypothetically the incommensurability of the diagonal, since a falsehood results through contradicting this. For this we found to be reasoning per impossibile, viz. proving something impossible by means of an hypothesis conceded at the beginning.
ὥστ᾽ ἐπεὶ τοῦ ψεύδους γίνεται συλλογισμὸς δεικτικὸς ἐν τοῖς εἰς τὸ ἀδύνατον ἀπαγομένοις, τὸ δ᾽ ἐξ ἀρχῆς ἐξ ὑποθέσεως δείκνυται, τοὺς δὲ δεικτικοὺς πρότερον εἴπομεν ὅτι διὰ τούτων περαίνονται τῶν σχημάτων, φανερὸν ὅτι καὶ οἱ διὰ τοῦ ἀδυνάτου συλλογισμοὶ διὰ τούτων ἔσονται τῶν σχημάτων.	Quare quoniam falsus fit syllogismus ostensivus in his quae ad impossibile deducuntur, quod autem est ex principio, ex hypothesi monstratur, ostensivos autem diximus prius, quoniam per has terminantur figuras, manifestum quoniam et per impossibile syllogismi per has erunt figuras.	Consequently, since the falsehood is established in reductions ad impossibile by an ostensive syllogism, and the original conclusion is proved hypothetically, and we have already stated that ostensive syllogisms are effected by means of these figures, it is evident that syllogisms per impossibile also will be made through these figures.
ὡσαύτως δὲ καὶ οἱ ἄλλοι πάντες οἱ ἐξ ὑποθέσεως· ἐν ἅπασι γὰρ ὁ μὲν συλλογισμὸς γίνεται πρὸς τὸ μεταλαμβανόμενον, τὸ δ᾽ ἐξ ἀρχῆς περαίνεται δι᾽ ὁμολογίας ἤ τινος ἄλλης ὑπο ↵ θέσεως. εἰ δὲ τοῦτ᾽ ἀληθές, πᾶσαν ἀπόδειξιν καὶ πάντα συλλογισμὸν ἀνάγκη γίνεσθαι διὰ τριῶν τῶν προειρημένων σχημάτων. τούτου δὲ δειχθέντος δῆλον ὡς ἅπας τε συλλογισμὸς ἐπιτελεῖται διὰ τοῦ πρώτου σχήματος καὶ ἀνά γεται εἰς τοὺς ἐν τούτωι καθόλου συλλογισμούς.	Similiter autem et alii omnes qui sunt ex hypothesi, in omnibus his enim syllogismus quidem fit ad transsumptum, quod autem est ex principio, terminatur per confessionem aut per aliquam aliam hypothesin. Si autem hoc verum, necesse est omnem demonstrationem et omnem syllogismum fieri per tres praedictas figuras. (0666C) Hoc autem ostenso, palam quoniam omnis syllogismus perficitur per primam figuram, et reducitur in huius universales syllogismos.	Likewise all the other hypothetical syllogisms: for in every case the syllogism leads up to the proposition that is substituted for the original thesis; but the original thesis is reached by means of a concession or some other hypothesis. But if this is true, every demonstration and every syllogism must be formed by means of the three figures mentioned above. But when this has been shown it is clear that every syllogism is perfected by means of the first figure and is reducible to the universal syllogisms in this figure.